In canonical coordinates, the explicit form of the flat metric \(\mathring{\mathbf {g}}\)  isThis is essentially eq (4.75) from [1]}, with a couple of typos corrected (in the most
up-to-date version as of writing) and \(\mathbf {\hat{e}}^{0}\)  used instead of its scaled version
\(\mathbf {\hat{\epsilon }}^{0}\) .
\( \mathring{\mathbf {g}} = \sum _{k = 1}^{n} \frac{X_{k}}{U_{k}} \bigg \vert _{M_{k} = 0} \mu ^{k}\mu ^{k} - i \sum _{k = 1}^{n} \left(\mu ^{k} dx_{k} + dx_{k}\ \mu ^{k}\right) +\varepsilon \ \mathbf {\hat{e}}^{0} \mathbf {\hat{e}}^{0}.\) 
